The Annals of Probability

Gaussian fluctuations for non-Hermitian random matrix ensembles

B. Rider and Jack W. Silverstein

Full-text: Open access


Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4+ɛ) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$ and letting N→∞, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type XN(f)=∑k=1Nf(λk) where λ1, λ2, …, λN denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533–605], where the analogous result for random sample covariance matrices is established.

Article information

Ann. Probab., Volume 34, Number 6 (2006), 2118-2143.

First available in Project Euclid: 13 February 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52
Secondary: 60F05: Central limit and other weak theorems

Random matrix theory central limit theorem spectral statistics


Rider, B.; Silverstein, Jack W. Gaussian fluctuations for non-Hermitian random matrix ensembles. Ann. Probab. 34 (2006), no. 6, 2118--2143. doi:10.1214/009117906000000403.

Export citation


  • Bai, Z. D. (1997). Circular law. Ann. Probab. 25 494--529.
  • Bai, Z. D. and Silverstein, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 316--345.
  • Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 533--605.
  • Bell, S. R. (1992). The Cauchy Transform, Potential Theory, and Conformal Mapping. CRC Press, Boca Raton, FL.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • Costin, O. and Lebowtz, J. (1995). Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75 69--72.
  • Diaconis, P. and Evans, S. N. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615--2633.
  • Edelman, A. (1997). The probability that a random real matrix has $k$ real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60 203--232.
  • Forrester, P. J. (1999). Fluctuation formula for complex random matrices. J. Phys. A 32 159--163.
  • Fyodorov, Y. V. and Sommers, H.-J. (2003). Random matrices close to Hermitian and unitary: Overview of methods and results. J. Phys. A 36 3303--3347.
  • Geman, S. (1980). A limit theorem for the norm of random matrices. Ann. Probab. 8 252--261.
  • Geman, S. (1986). The spectral radius of large random matrices. Ann. Probab. 14 1318--1328.
  • Girko, V. L. (1984). Circle law. Theory Probab. Appl. 29 694--706.
  • Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 440--449.
  • Guionnet, A. (2002). Large deviations and upper bounds for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré Probab. Statist. 38 341--384.
  • Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge Univ. Press.
  • Hughes, C. P., Keating, J. P. and O'Connell, N. (2000). On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys. 220 429--451.
  • Hwang, C. R. (1986). A brief survey on the spectral radius and the spectral distribution of large dimensional random matrices with iid entries. In Random Matrices and Their Applications (M. L. Mehta, ed.) 50 145--152. Amer. Math. Soc., Providence, RI.
  • Israelson, S. (2001). Asymptotic fluctuations of a particle system with singular interaction. Stochastic Process. Appl. 93 25--56.
  • Keating, J. P. and Snaith, N. C. (2000). Random matrix theory and $\zeta(1/2 + i t)$. Comm. Math. Phys. 214 57--89.
  • Johansson, K. (1997). On random matrices from the classical compact groups. Ann. of Math. (2) 145 519--545.
  • Johansson, K. (1998). On the fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151--204.
  • Lebœuf, P. (1999). Random matrices, random polynomials and coulomb systems. In Proceedings of the International Conference on Strongly Coupled Coulomb Systems, Saint-Malo.
  • Peres, Y. and Vir$\acute\mboxa$g, B. (2005). Zeros of the i.i.d. Gaussian power series: A conformally invariant determinantal process. Acta Math. 194 1--35.
  • Rider, B. (2004). Deviations from the circular law. Probab. Theory Related Fields 130 337--367.
  • Sinai, Ya. and Soshnikov, A. (1998). Central limit theorems for traces of large random matrices with independent entries. Bol. Soc. Brasil. Mat. 29 1--24.
  • Soshnikov, A. (2002). Gaussian limits for determinantal random point fields. Ann. Probab. 30 171--181.
  • Titchmarsh, E. C. (1939). The Theory of Functions, 2nd ed. Oxford Univ. Press.
  • Wieand, K. (2002). Eigenvalue distributions of random unitary matrices. Probab. Theory Related Fields 123 202--224.
  • Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. (1988). On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Related Fields 78 509--521.